Problem: $\dfrac{ -3k - 7l }{ -9 } = \dfrac{ -k - 7m }{ 10 }$ Solve for $k$.
Answer: Multiply both sides by the left denominator. $\dfrac{ -3k - 7l }{ -{9} } = \dfrac{ -k - 7m }{ 10 }$ $-{9} \cdot \dfrac{ -3k - 7l }{ -{9} } = -{9} \cdot \dfrac{ -k - 7m }{ 10 }$ $-3k - 7l = -{9} \cdot \dfrac { -k - 7m }{ 10 }$ Multiply both sides by the right denominator. $-3k - 7l = -9 \cdot \dfrac{ -k - 7m }{ {10} }$ ${10} \cdot \left( -3k - 7l \right) = {10} \cdot -9 \cdot \dfrac{ -k - 7m }{ {10} }$ ${10} \cdot \left( -3k - 7l \right) = -9 \cdot \left( -k - 7m \right)$ Distribute both sides ${10} \cdot \left( -3k - 7l \right) = -{9} \cdot \left( -k - 7m \right)$ $-{30}k - {70}l = {9}k + {63}m$ Combine $k$ terms on the left. $-{30k} - 70l = {9k} + 63m$ $-{39k} - 70l = 63m$ Move the $l$ term to the right. $-39k - {70l} = 63m$ $-39k = 63m + {70l}$ Isolate $k$ by dividing both sides by its coefficient. $-{39}k = 63m + 70l$ $k = \dfrac{ 63m + 70l }{ -{39} }$ Swap signs so the denominator isn't negative. $k = \dfrac{ -{63}m - {70}l }{ {39} }$